# Difference between revisions of "Discontinuous derivative"

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− | Consider the function | + | Consider the function (blue curve) |

:<math> f: \mathbb{R} \to \mathbb{R}, x \mapsto | :<math> f: \mathbb{R} \to \mathbb{R}, x \mapsto | ||

\begin{cases} | \begin{cases} | ||

Line 7: | Line 7: | ||

</math> | </math> | ||

<math>f</math> is a continous and differentiable. | <math>f</math> is a continous and differentiable. | ||

− | The derivative of <math>f</math> is the function | + | The derivative of <math>f</math> is the function (red curve) |

:<math> | :<math> | ||

f': \mathbb{R} \to \mathbb{R}, x \mapsto | f': \mathbb{R} \to \mathbb{R}, x \mapsto | ||

Line 15: | Line 15: | ||

\end{cases}\,. | \end{cases}\,. | ||

</math> | </math> | ||

− | We observe that <math>f'(0) = 0</math> but <math>\lim_{x\to0}f'(x)</math> does not exist. | + | We observe that <math>f'(0) = 0</math>, but <math>\lim_{x\to0}f'(x)</math> does not exist. |

Therefore, <math>f'</math> is an example of a derivative which is not continuous. | Therefore, <math>f'</math> is an example of a derivative which is not continuous. |

## Revision as of 11:02, 13 February 2019

Consider the function (blue curve)

- [math] f: \mathbb{R} \to \mathbb{R}, x \mapsto \begin{cases} x^2\sin(1/x),& x\neq 0\\ 0,& x=0 \end{cases}\,. [/math]

[math]f[/math] is a continous and differentiable. The derivative of [math]f[/math] is the function (red curve)

- [math] f': \mathbb{R} \to \mathbb{R}, x \mapsto \begin{cases} 2\sin(1/x) - \cos(1/x), &x \neq 0\\ 0,& x=0 \end{cases}\,. [/math]

We observe that [math]f'(0) = 0[/math], but [math]\lim_{x\to0}f'(x)[/math] does not exist.

Therefore, [math]f'[/math] is an example of a derivative which is not continuous.

### The underlying JavaScript code

```
var board = JXG.JSXGraph.initBoard('jxgbox', {axis:true, boundingbox:[-1/2,1/2,1/2,-1/2]});
var g = board.create('functiongraph', ["2*sin(1/x) - cos(1/x)"], {strokeColor: 'red'});
var f = board.create('functiongraph', ["x^2*sin(1/x)"], {strokeWidth:2});
```